NAG CL Interface
f08ygc (dtgsen)

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1 Purpose

f08ygc reorders the generalized Schur factorization of a matrix pair in real generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements, or blocks on the diagonal of the generalized Schur form. The function also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.

2 Specification

#include <nag.h>
void  f08ygc (Nag_OrderType order, Integer ijob, Nag_Boolean wantq, Nag_Boolean wantz, const Nag_Boolean select[], Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double q[], Integer pdq, double z[], Integer pdz, Integer *m, double *pl, double *pr, double dif[], NagError *fail)
The function may be called by the names: f08ygc, nag_lapackeig_dtgsen or nag_dtgsen.

3 Description

f08ygc factorizes the generalized real n×n matrix pair (S,T) in real generalized Schur form, using an orthogonal equivalence transformation as
S = Q^ S^ Z^T ,   T= Q^ T^ Z^T ,  
where (S^,T^) are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. The leading columns of Q and Z are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair (S,T).
The pair (S,T) are in real generalized Schur form if S is block upper triangular with 1×1 and 2×2 diagonal blocks and T is upper triangular as returned, for example, by f08xcc, or f08xec with job=Nag_Schur. The diagonal elements, or blocks, define the generalized eigenvalues (αi,βi), for i=1,2,,n, of the pair (S,T). The eigenvalues are given by
λi = αi / βi ,  
but are returned as the pair (αi,βi) in order to avoid possible overflow in computing λi. Optionally, the function returns reciprocals of condition number estimates for the selected eigenvalue cluster, p and q, the right and left projection norms, and of deflating subspaces, Difu and Difl. For more information see Sections 2.4.8 and 4.11 of Anderson et al. (1999).
If S and T are the result of a generalized Schur factorization of a matrix pair (A,B)
A = QSZT ,   B= QTZT  
then, optionally, the matrices Q and Z can be updated as QQ^ and ZZ^. Note that the condition numbers of the pair (S,T) are the same as those of the pair (A,B).

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: ijob Integer Input
On entry: specifies whether condition numbers are required for the cluster of eigenvalues (p and q) or the deflating subspaces (Difu and Difl).
ijob=0
Only reorder with respect to select. No extras.
ijob=1
Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster (p and q).
ijob=2
The upper bounds on Difu and Difl. F-norm-based estimate (stored in dif[0] and dif[1] respectively).
ijob=3
Estimate of Difu and Difl. 1-norm-based estimate (stored in dif[0] and dif[1] respectively). About five times as expensive as ijob=2.
ijob=4
Compute pl, pr and dif as in ijob=0, 1 and 2. Economic version to get it all.
ijob=5
Compute pl, pr and dif as in ijob=0, 1 and 3.
Constraint: 0ijob5.
3: wantq Nag_Boolean Input
On entry: if wantq=Nag_TRUE, update the left transformation matrix Q.
If wantq=Nag_FALSE, do not update Q.
4: wantz Nag_Boolean Input
On entry: if wantz=Nag_TRUE, update the right transformation matrix Z.
If wantz=Nag_FALSE, do not update Z.
5: select[n] const Nag_Boolean Input
On entry: specifies the eigenvalues in the selected cluster. To select a real eigenvalue λj, select[j-1] must be set to Nag_TRUE.
To select a complex conjugate pair of eigenvalues λj and λj+1, corresponding to a 2×2 diagonal block, either select[j-1] or select[j] or both must be set to Nag_TRUE; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.
6: n Integer Input
On entry: n, the order of the matrices S and T.
Constraint: n0.
7: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the matrix S in the pair (S,T).
On exit: the updated matrix S^.
8: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
9: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least max(1,pdb×n).
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the matrix T, in the pair (S,T).
On exit: the updated matrix T^
10: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax(1,n).
11: alphar[n] double Output
On exit: see the description of beta.
12: alphai[n] double Output
On exit: see the description of beta.
13: beta[n] double Output
On exit: alphar[j-1] / beta[j-1] and alphai[j-1] / beta[j-1] are the real and imaginary parts respectively of the jth eigenvalue, for j=1,2,,n.
If alphai[j-1] is zero, then the jth eigenvalue is real; if positive then alphai[j] is negative, and the jth and (j+1)st eigenvalues are a complex conjugate pair.
Conjugate pairs of eigenvalues correspond to the 2×2 diagonal blocks of S^. These 2×2 blocks can be reduced by applying complex unitary transformations to (S^,T^) to obtain the complex Schur form (S~,T~) , where S~ is triangular (and complex). In this form alphar+ialphai and beta are the diagonals of S~ and T~ respectively.
14: q[dim] double Input/Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when wantq=Nag_TRUE;
  • 1 otherwise.
The (i,j)th element of the matrix Q is stored in
  • q[(j-1)×pdq+i-1] when order=Nag_ColMajor;
  • q[(i-1)×pdq+j-1] when order=Nag_RowMajor.
On entry: if wantq=Nag_TRUE, the n×n matrix Q.
On exit: if wantq=Nag_TRUE, the updated matrix QQ^.
If wantq=Nag_FALSE, q is not referenced.
15: pdq Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if wantq=Nag_TRUE, pdq max(1,n) ;
  • otherwise pdq1.
16: z[dim] double Input/Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when wantz=Nag_TRUE;
  • 1 otherwise.
The (i,j)th element of the matrix Z is stored in
  • z[(j-1)×pdz+i-1] when order=Nag_ColMajor;
  • z[(i-1)×pdz+j-1] when order=Nag_RowMajor.
On entry: if wantz=Nag_TRUE, the n×n matrix Z.
On exit: if wantz=Nag_TRUE, the updated matrix ZZ^.
If wantz=Nag_FALSE, z is not referenced.
17: pdz Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if wantz=Nag_TRUE, pdz max(1,n) ;
  • otherwise pdz1.
18: m Integer * Output
On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).
19: pl double * Output
20: pr double * Output
On exit: if ijob=1, 4 or 5, pl and pr are lower bounds on the reciprocal of the norm of ‘projections’ p and q onto left and right eigenspaces with respect to the selected cluster. 0<pl, pr1.
If m=0 or m=n, pl=pr=1.
If ijob=0, 2 or 3, pl and pr are not referenced.
21: dif[dim] double Output
Note: the dimension, dim, of the array dif must be at least 2.
On exit: if ijob2, dif[0] and dif[1] store the estimates of Difu and Difl.
If ijob=2 or 4, dif[0] and dif[1] are F-norm-based upper bounds on Difu and Difl.
If ijob=3 or 5, dif[0] and dif[1] are 1-norm-based estimates of Difu and Difl.
If m=0 or n, dif[0] and dif[1] =(A,B)F.
If ijob=0 or 1, dif is not referenced.
22: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONSTRAINT
Constraint: if wantq=Nag_TRUE, pdq max(1,n) ;
otherwise pdq1.
Constraint: if wantz=Nag_TRUE, pdz max(1,n) ;
otherwise pdz1.
NE_INT
On entry, ijob=value.
Constraint: 0ijob5.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SCHUR
Reordering of (S,T) failed because the transformed matrix pair would be too far from generalized Schur form; the problem is very ill-conditioned. (S,T) may have been partially reordered. If requested, 0 is returned in dif[0] and dif[1], pl and pr.

7 Accuracy

The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices (S+E) and (T+F), where
E2 = Oε S2   and   F2= Oε T2 ,  
and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem, and for information on the condition numbers returned.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08ygc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this function is f08yuc.

10 Example

This example reorders the generalized Schur factors S and T and update the matrices Q and Z given by
S = ( 4.0 1.0 1.0 2.0 0.0 3.0 4.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 6.0 ) ,   T= ( 2.0 1.0 1.0 3.0 0.0 1.0 2.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 2.0 ) ,  
Q = ( 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 )   and   Z= ( 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 ) ,  
selecting the first and fourth generalized eigenvalues to be moved to the leading positions. Bases for the left and right deflating subspaces, and estimates of the condition numbers for the eigenvalues and Frobenius norm based bounds on the condition numbers for the deflating subspaces are also output.

10.1 Program Text

Program Text (f08ygce.c)

10.2 Program Data

Program Data (f08ygce.d)

10.3 Program Results

Program Results (f08ygce.r)